Counting triangles in power-law uniform random graphs

Pu Gao¹ ; Remco van der Hofstad ; Angus Southwell ; Clara Stegehuis

¹School of MathematicsMonash UniversityAustralia

The electronic journal of combinatorics, Tome 27 (2020) no. 3

Résumé

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $\tau\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the probabilities that edges are present are moderated by asymptotically linear functions of the products of these vertex weights.

Détail

DOI : 10.37236/9239

Classification : 05C80, 05C30
Mots-clés : asymptotic number of triangles in uniform random graphs